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Solve the system of equations.\newliney=x2−50x−50y = x^2 - 50x - 50\newliney=−50x+31y = -50x + 31\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)

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Solve a system of linear and quadratic equations: parabolasright chevron icon

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Q. Solve the system of equations.\newliney=x2−50x−50y = x^2 - 50x - 50\newliney=−50x+31y = -50x + 31\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)
  1. Set Equations Equal: We have the following system of equations:\newliney=x2−50x−50y = x^2 - 50x - 50\newliney=−50x+31y = -50x + 31\newlineTo find the solution, we will set the two equations equal to each other since they both equal yy.\newlinex2−50x−50=−50x+31x^2 - 50x - 50 = -50x + 31
  2. Simplify Quadratic Equation: Simplify the equation by moving all terms to one side to form a quadratic equation.\newlinex2−50x−50+50x−31=−50x+31+50x−31x^2 - 50x - 50 + 50x - 31 = -50x + 31 + 50x - 31\newlinex2−81=0x^2 - 81 = 0
  3. Solve for x: Solve the quadratic equation for x.\newlinex2=81x^2 = 81\newlineTake the square root of both sides.\newlinex=±81x = \pm\sqrt{81}\newlinex=±9x = \pm9
  4. Find y-values: Find the corresponding y-values for each xx-value by substituting back into one of the original equations. We can use y=−50x+31y = -50x + 31. For x=9x = 9: y=−50(9)+31y = -50(9) + 31 y=−450+31y = -450 + 31 y=−419y = -419
  5. Find Second y-value: Find the y-value for the second x-value.\newlineFor x=−9x = -9:\newliney=−50(−9)+31y = -50(-9) + 31\newliney=450+31y = 450 + 31\newliney=481y = 481
  6. Write Coordinates: Write the coordinates in exact form.\newlineThe solutions to the system of equations are the points where the two graphs intersect.\newlineFirst Coordinate: (9,−419)(9, -419)\newlineSecond Coordinate: (−9,481)(-9, 481)

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